2 Light Scattering and Absorption by Nonspherical Ice Crystalsatmos.ucla.edu/.../Yang_Chap2_Springer_2006.pdf · principles of geometric optics in terms of the ray-tracing technique

2 Light Scattering and Absorptionby Nonspherical Ice Crystals

Ping Yang and Kuo-Nan Liou

2.1 Introduction

The majority of ice crystals in the atmosphere exist in cirrus clouds, cloudsthat normally reside in the upper troposphere in midlatitudes. In the tropics,ice clouds associated with deep cumulus convections (Houze, 1993) can extendto the lower stratosphere. Ice crystals have also been frequently observed in thepolar regions because of low temperatures. The global cirrus cover has beenestimated to be about 20% to 25%, but recent analysis using the 15-µm CO2satellite channels has shown that their occurrence frequency can be larger than50% in the tropics (Wylie et al., 1994). The inclusion of the 1.375-µm water vaporabsorption channel (Gao and Kaufman, 1995) in the recent Moderate ResolutionImaging Spectroradiometer (MODIS) instrument on Terra and Acqua satelliteplatforms (King et al. 2003) has offered an unprecedented opportunity to detectoptically thin cirrus. However, many thin and subvisual cirrus clouds could havebeen missed from the implementation of various past and current passive satellitedetection techniques.

From analysis of the MODIS images acquired from the visible and 1.375-µmchannels, it has been shown that most clear-sky pixels identified by the visiblechannels actually contain thin cirrus (Roskovensky and Liou, 2003; Dessler andYang, 2003; Meyer et al., 2004). Because of their high location in the atmo-sphere and the complex microphysical properties of ice crystals within them,cirrus clouds differ significantly from low and middle clouds in terms of their ra-diative properties. High cirrus clouds reflect a portion of the incoming sunlight,referred to as the solar albedo effect. But at the same time these clouds can alsoeffectively trap a significant amount of the thermal infrared radiation emittedfrom the surface and lower atmosphere, referred to as the infrared greenhouseeffect (Liou, 1992). The intrinsic radiative properties of cirrus clouds determinethe competition between the solar albedo and infrared greenhouse effects (Liou,1986; Stephens et al., 1990), essential to the discussion of the Earth’s climate andclimate change. Moreover, cirrus clouds are closely related to the water vapordistribution near the upper troposphere and the lower stratosphere (Jensen etal., 1996; Holton and Gettelman, 2001). The important roles of cirrus clouds in

32 Ping Yang and Kuo-Nan Liou

various atmospheric processes have been discussed by Liou (1986, 1992) as wellas in a number of recent studies (see Lynch et al., 2002).

Ice crystals in cirrus clouds are almost exclusively nonspherical particles (e.g.,Heymsfield and Iaquinta, 2000), ranging from plates, solid and hollow columns,bullet rosettes, and aggregates, to more irregular shapes with various complexsurface morphological conditions (e.g., surface roughness). The effect of non-spherical ice crystals within cirrus clouds on their bulk radiative properties ispronounced and must be accounted for in the development of remote sensingtechniques and climate analysis. Liou et al. (2000) demonstrated that the ap-proximation of nonspherical ice crystals as equivalent ice spheres for the single-scattering and radiative transfer processes can substantially underestimate thealbedo of the cirrus. Moreover, the single-scattering properties associated withproper ice crystal morphologies must be used for a correct interpretation ofother bulk optical properties of cirrus clouds, particularly, the polarization con-figuration (Liou and Takano, 2002). Mishchenko et al. (1996), Rolland et al.(2000), and Yang et al. (2001a) also showed strong sensitivity of the cirrus cloudalbedo, bidirectional reflectance, and the accuracy of optical thickness retrievalto the ice particle shape assumed. Consequently, it is critically important thatthe nonsphericity of ice crystals be accurately modeled in radiative transfer com-putations and remote sensing implementations involving cirrus clouds.

Because of the importance of ubiquitous cirrus clouds in remote sensing andclimate research, substantial efforts have been made in the last three decadesto understand and determine the fundamental scattering and absorption prop-erties of ice crystals. Early research efforts to account for the nonsphericity ofice crystals in cirrus can be traced back to the studies by Liou (1972a, 1972b)and Stephens (1980a, 1980b) who assumed that these clouds are composed oflong circular cylinders. The analytical solution for the scattering of light by aninfinite circular cylinder at normal incidence was developed by Lord Rayleigh(1918). Wait (1955), Kerker (1969), and Liou (1972a) extended the solutionsfor oblique incidence. The single-scattering properties of ice crystals assumingcircular-cylinder shape, however, cannot be used to explain a number of opti-cal phenomena associated with cirrus clouds, for example, the well-known 22◦

halo. The simplest habits (or shapes) of realistic ice crystals are columns andplates with well-defined hexagonal structures. Even for this type of ice crys-tal, it appears not to be possible to impose an appropriate coordinate systemto analytically solve the associated electromagnetic wave equation subjected tothe boundary conditions at the surface of a hexagonal particle. The difficultyin conjunction with the application of the variable separation method to com-plex ice crystal morphologies (e.g., bullet rosettes and aggregates) would bemore prominent. This is because proper coordinate systems for the variable sep-aration method can only be defined in cases involving the scattering of lightby spheres (Lorenz, 1890; Mie, 1908), spheroids (Oguchi, 1973; Asano and Ya-mamoto, 1975), and infinite cylinders (Kerker, 1969; Liou, 1972a).

From the late 1970s to the 1990s, the geometric optics method by meansof the ray-tracing technique has been extensively used to investigate the single-scattering properties of nonspherical ice crystals (Wendling et al., 1979; Coleman

2 Light Scattering and Absorption by Nonspherical Ice Crystals 33

and Liou, 1981; Cai and Liou, 1982; Takano and Jayaweera, 1985; Takano andLiou, 1989a, 1989b; Macke, 1993; Hess and Wiegner, 1994; Macke et al., 1996a).Note that application of the ray-tracing technique to light scattering by a sphereand a hexagonal column can be traced back to the study by Liou and Hansen(1971) and Jacobowitz (1971), respectively. The research results from these ef-forts have been used in various applications in conjunction with the study ofcirrus clouds, ranging from remote sensing (e.g. Minnis et al., 1993a, 1993b) tothe parameterization of the radiative properties of ice clouds (Ebert and Curry,1992; Fu and Liou, 1993) for use in climate models.

Recent reviews by Wriedt (1998), Kokhanovsky (1999), Mishchenko et al.(2000), Liou (2002), and Kahnert (2003) have enumerated various methods thathave been developed for the solution of light scattering by nonspherical particles.These include the T-matrix method (Waterman, 1971; Mishchenko and Travis,1998), the discrete dipole approximation (Purcell and Pennypacker, 1973; Draineand Flatau, 1994), the finite-difference time domain (FDTD) method (Yee,1966; Taflove and Brodwin, 1975; Holland, 1977; Kunz and Lee, 1978; Taflove,1980; Kunz and Simpson, 1981; Umashankar and Taflove, 1982; Taflove, andUmashankar, 1990; Yang and Liou, 1996a; Yang et al., 2000; Sun et al., 1999),and the boundary-element method (Miller, 1988; Kress, 1990; Mano, 2000), andvarious approximate methods including the geometric optics approximation. Ap-plications of some of these methods to light scattering by ice crystals have beenshown to be useful for a better understanding of the optical and radiative prop-erties of cirrus clouds.

In this chapter, we review the progress in the studies of the scattering andabsorption properties of nonspherical ice crystals in the Earth’s atmosphere fromthe theoretical and computational perspectives. Specifically, we will highlight themethod of geometric optics and the relevant improvements based on which ray-tracing can be performed for large ice crystals with complex geometries and theFDTD numerical method that can be used for the solution of light scatteringby small nonspherical and inhomogeneous ice crystals. No originality is claimedin this review; however, we have made an effort to systematically recapture thegeometric optics approach and illustrate the basic concepts of the FDTD methodand the major numerical steps associated with its application to light scatteringand absorption by ice crystals.

2.2 Geometric Optics for Light Scatteringby Large Ice Crystals

According to aircraft observations (e.g., Mitchell et al., 1996; McFarquhar et al.,1999) by means of the optical imaging, high resolution video camera, and repli-cator techniques, the ice crystal size distribution in various types of cirrus cloudsranges from about 10 micrometers to thousands of micrometers. For visible andnear-infrared wavelengths, the size parameters (defined as πD/λ in which D andλ are the particle’s characteristic dimension and the incident wavelength, respec-tively) associated with these particles are large enough that we may apply the


principles of geometric optics in terms of the ray-tracing technique. van de Hulst(1957) and Liou and Hansen (1971) applied this technique to light scattering byspheres. The latter authors also compared the phase function and polarizationpatterns computed from the geometric ray-tracing method and the Lorenz–Mietheory for polydisperse spheres. The solution from the ray-tracing technique isreasonably accurate when the modal size parameter is larger than 100 for apolydisperse system of spheres. The earliest application of ray-tracing to lightscattering by hexagonal ice prisms was first carried out by Jacobowitz (1971). Inhis study, a sufficiently large number of equally spaced, parallel rays were tracedthrough a hexagonal ice crystal. The external reflection and two refractions aftervarious orders of internal reflections were summed to determine the scatteringpattern in the far field. Diffraction contribution to the scattering of the incidentradiation in the forward direction can be computed by Kirchhoff’s formula. InJacobowitz’s study, hexagonal prisms were assumed to be infinite by long, whichis not realistic. To circumvent this shortcoming, Wendling et al. (1979) combinedthe Monte Carlo method and the ray-tracing technique to compute the phasefunction of finite hexagonal columns. In these early studies, the polarization ef-fect and phase interferences associated with the incident, internal, and scatteredrays were not accounted for in the calculation.

Cai and Liou (1982) were the first to include the polarization configurationand phase interferences in ray-tracing to compute the single-scattering proper-ties of hexagonal columns and plates. The theoretical foundation for the ‘con-ventional’ geometric ray tracing and the associated computional algorithm de-veloped by Cai and Liou (1982) were later improved and refined by a numberof researchers. Takano and Liou (1989a, 1989b) considered the effects of the icecrystal’s birefringence property, horizontal orientation, and size spectrum in as-sociation with light scattering calculations. The single-scattering properties ofice crystals with horizontally orientation have also been investigated by Rock-witz (1989) and Noel et al. (2001). The ray-tracing method has been applied tovarious complex ice crystal shapes by Takano and Liou (1995), Macke (1993),Macke et al. (1996a), Iaquinta et al. (1995), Muinonen et al. (1997), Peltoniemiet al. (1998), and Yang and Liou (1998), and to ice crystals with inclusions (e.g.,air bubbles and soots) by Macke et al. (1996b) and Macke (2000). Applicationof the ray-tracing technique implemented with the Monte Carlo method to com-plex geometries have also been recently reported by Nousiainen et al. (2003),and Grynko and Shkuratov (2003). Most recently, Borovoi et al. (2002) inves-tigated the scattering characteristics (backscattering features, in particular) ofhexagonal ice crystals with arbitrary orientations using the ray-tracing method.Additionally, Borovoi and Grishin (2003) reported an effective ray-tracing algo-rithm for computing the Jones scattering matrix, and subsequently, the Muellermatrix.

Alternate approaches to the conventional ray-tracing method have been de-veloped by Muinonen (1989) and Yang and Liou (1995, 1996b, and 1997) inwhich the principles of geometric optics are applied to the computation of thenearfield either on the surface of or inside the scattering particle. The near fieldobtained from the ray-tracing technique is then mapped to the far field on the


basis of either a surface-integral- or a volume-integral-based electromagnetic re-lation. Below we systematically recast the theoretical basis of the conventionalray-tracing technique, and follow with a concise presentation of two versions ofthe improved geometric optics method for the scattering of light by ice crystals.

The ray-tracing algorithm for the scattering of polarized light by nonsphericalice crystals that was formulated by Cai and Liou (1982) used various specificcoordinate systems. To simplify the formulation presented in this paper, we shalladopt a vector form (Yang and Cai, 1990; Yang and Liou, 1996b, 1997) that isindependent of specific coordinate systems. As shown in Fig. 2.1, when the sizeparameter associated with the scattering particle is large, the incident field canbe thought of as composing of a bundle of localized waves or rays. As articulatedin Cai and Liou (1982), the width of localized rays must be much larger thanthe incident wavelength and yet smaller than the ice crystal size.

Consider an incident ray that passes through point Q0 and propagates alongthe incident direction specified by a unit vector ei0. The ray first impinges onthe particle surface at point Q1 where external reflection and refraction occur.For reflection and refraction at this incident point, the directions of the incident,reflected, and refracted rays are denoted by the unit vectors ei1, e

r1, and et1, re-

Fig. 2.1. A conceptual diagram for the principle of the ray-tracing technique forcomputing the single-scattering properties of a particle that is much larger than theincident wavelength.


spectively. The first-order refracted ray then impinges on the next incident pointat Q2 where the first-order internal reflection and the corresponding refractionoccur. Note that for the incidence at the point Q2, the incident direction ei2 isthe same as that of the first-order refracted ray. Likewise, the subsequent inter-nal reflections and refractions occur at the points Qp in which the order of theray, p, is larger than 2. The tracing of a ray can be terminated when the energycarried by this ray is practically negligible.

To trace the reflected and refracted rays, let np (p = 1, 2, 3, . . . ) be the unitvectors locally normal to the particle surfaces at the incident points Qp (p = 1, 2,3, . . . ) facing the incoming rays, as shown in Fig. 2.2. For the external reflectionat the point Q1, the incident direction ei1 and the incident angle θi

1 are given,respectively, by the following two expressions:

ei1 = ei0, (2.1)θi1 = cos−1(−n1 · ei1), (2.2)

where ei0 denotes the initial incident direction (Fig. 2.1). Following Snell’s law,the directions of the externally reflected ray and the corresponding refracted rayare defined by

er1 = ei1 + 2 cos θi1n1, (2.3)

et1 = ei1/m+ (cos θi1/m− cos θt

1)n1, (2.4)

where m is the refractive index of the scattering particle, and θt1 is the refractive

angle given by Snell’s law as follows:

θt1 = sin−1(sin θi

1/m). (2.5)

When the refractive index, m, is a complex number, simultaneous absorptionand scattering occur and the refracted wave within the particle is an inhomo-geneous wave (Born and Wolf, 1959; Bohren and Huffman, 1983). In this case,

Fig. 2.2. Schematic diagrams for the directions of the incident, reflected and refractedrays. Also shown are the unit vectors for specifying the polarization configuration. Notethat v1 and vp (not shown in the diagram) point out of the paper.


an adjusted refractive must be used to trace the refracted rays (Yang and Liou,1995). Note that Born and Wolf (1959) only formulated the Fresnel formulaby avoiding the complex refraction angle when absorption occurs. Yang et al.(2001b) showed that in general the electric field vector associated with the re-fracted rays may not be perpendicular to the ray direction and developed animproved scheme for the ray-tracing computation. For practical computationsat the visible and near-infrared wavelengths, the real part of the refractive indexmay be used as an excellent approximation of the adjusted refractive index fortracing the ray directions on the basis of eqs (2.4) and (2.5). The issue associatedwith the inhomogeneous wave properties within an ice crystal involving complexrefractive index will not be elaborated further. Interested readers in this subjectmay wish to consult with the work of Born and Wolf (1959), Bohren and Huff-man (1983), Dupertuis et al. (1994), Yang and Liou (1995), Yang et al. (2001b),Liou (2002), and Chang et al. (2005).

For internal reflection with orders of p = 2, 3, 4, . . . , the incident directionscan be defined in a likely manner and are given by the directions of either thefirst-order refracted rays or internally reflected rays as follows:

ei2 = et1, (2.6)eip = erp−1, p = 3, 4, 5, . . . (2.7)

With some vector algebraic manipulations on the basis of Snell’s law, it can beshown that the propagating directions of the pth order reflected and refractedrays are given by

erp = eip + 2 cos θipnp, (2.8)

etp = meip + (m cos θip − cos θt

p)np, (2.9)

where the incident and refraction angles, θip and θt

p, are defined via the followingexpressions:

θip = cos−1(−np · eip), (2.10)

θtp = sin−1(m sin θi

p). (2.11)

The total reflection occurs if the term m sin θip in eq. (2.11) is larger than 1. In

this case, a refracted ray should not be expected and the ray-tracing computationshould be continued only for the ray associated with total reflection. Equations(2.1)–(2.11) constitute a closed set of equations for tracing the directions of allthe reflected and refracted rays associated with a given incident ray.

A localized plane electromagnetic wave is a transverse vector wave. Thus,the vector property or the polarization configuration of the electric fields associ-ated with localized rays in the ray-tracing computation must be accounted for.To include the polarization configuration, we shall define various auxiliary unitvectors. For the incident direction of an initial ray specified by a unit vector ei0,we define two unit vector u0 and v0 (see Fig. 2.1) that are normal to the incidentdirection and satisfy the relations as follows:


u0 · v0 = 0 and v0 × u0 = ei0. (2.12)

The unit vectors v0, u0 and ei0 defined in this equation constitute a right-handedcoordinate system. To define the initial rays in practice, we may specify theunit vectors v0, u0 and ei0 to point to the directions along the x-, y- and z-axisof the incident coordinate system. Because the unit vectors ei0, u0 and v0 areorthogonal to each other, the incident polarization configuration can be specifiedwith respect to u0 and v0, that is, the incident electric field Ei

0 can be writtenas follows:

Ei0 = Ei

0uu0 + Ei0v v0. (2.13)

Similarly, we may define three pairs of unit vectors, (uip, vp), (ur

p, vp), and (utp, vp)

for the ray directions along eip, erp and etp (p = 1, 2, 3, . . . ), respectively. The unit

vectors uip, u

rp and ut

p point along the directions shown in Fig. 2.2, if the unitvectors, vp, p = 1, 2, 3, . . . , (not shown in Fig. 2.2) are defined as being pointedout of the paper. These vectors can be specified via the following expressions:

vp = (eip × np)/ sin θip, p = 1, 2, 3, . . . , (2.14)

ui,r,tp = ei,r,tp × vp, p = 1, 2, 3, . . . .(2.15)

Evidently, the unit vector vp is normal to the incident plane, the plane containingthe incident direction and the direction locally normal to the particle surface atthe incident point, for the pth-order reflection and refraction. The unit vectorsui

p, urp, and ut

p are parallel to the pth-order incident plane. Note that vp in eq.(2.14) cannot be uniquely specified if sin θi

p = 0. In this case, we select vp = vp−1.With the aforementioned unit vectors defined, the electric fields associated withthe pth-order incident, reflected, and refracted rays can be expressed as follows:

Ei,r,tp = Ei,r,t

pu ui,r,tp + Ei,r,t

pv vp. (2.16)

Consider now the external reflection and the first-order refraction. In orderto apply the Fresnel formulas, the electric field associated with the incident rayimpinging on the point Q1 must be specified with respect to ui

1 and v1. Also,the electric field associated with the incident ray specified in eq. (2.13) can alsobe expressed in an alternative form as follows:

Ei0 = Ei

1uui1 + Ei

1v v1. (2.17)

Equations (2.13) and (2.17) for the incident electric vector are related via arotational matrix in the form

(Ei

1u

Ei1v

)= Γ1

(Ei

0u

Ei0v

), (2.18)

where Γ1 is a rotational matrix given by

Γ1 =(ui

1 · u0 ui1 · v0

v1 · u0 v1 · v0

). (2.19)


Because the field components on the left-hand side of eq. (2.18) are specifiedwith respect to the incident plane, the Fresnel formulas can be applied. Theexternally reflected field (Er

1) is given by

Er1 = Er

1uur1 + Er

1v v1, (2.20)

where (Er

1u

Er1v

)= R1

(Ei

1u

Ei1v

)= R1Γ1

(Ei

0u

Ei0v

). (2.21)

In eq. (2.21), R1 is the reflection matrix for the external reflection given by

R1 =(R1u 00 R1v

). (2.22)

The elements of the reflection matrix in eq. ( 2.22) are given by the Fresnelcoefficients (Born and Wolf, 1959) as follows:

R1u =m cos θi

1 − cos θt1

m cos θi1 + cos θt

1, (2.23)

R1v =cos θi

1 −m cos θt1

cos θi1 +m cos θt

1. (2.24)

Similarly, the electric field associated with the first-order refracted ray is givenby

Et1 = Et

1uut1 + Et

1v v1, (2.25)(Et

1u

Et1v

)= T1

(Ei

1u

Ei1v

)= T1Γ1

(Ei

0u

Ei0v

), (2.26)

where the refraction matrix T1 is defined in the form

T1 =(T1u 00 T1v

)=(

(1−R21u)1/2 0

0 (1−R21v)1/2

). (2.27)

In eq. (2.27), the conservation of the energy carried out by the ray due to achange in the refractive index and ray cross-section in two media is accountedfor in the refraction matrix (see eq. (48) in Cai and Liou (1982) and referencescited therein).

For the external reflection, the direction along the reflected ray is the scat-tering direction. Thus, the scattering angle is given by

θs1 = cos−1(ei0 · er1). (2.28)

The direction that is perpendicular to the scattering plane can be specified by

vs1 = ei0 × er1/ sin θs

1. (2.29)

If sin θs1 = 0 in eq. (2.29), implying the forward (i.e., θs

1 = 0◦) and backwardscattering (i.e., θs

1 = 180◦), the vector vs1 cannot be defined. In this case, we


select vs1 = v0. After the unit vector vs

1 is defined, the direction parallel to thescattering plane is given by

us1 = er1 × vs

1. (2.30)

If we express the electric field associated with the externally reflected ray withrespect to two directions parallel and perpendicular to the scattering plane, wehave

Er1 = Es

1uus1 + Es

1v vs1. (2.31)

From eqs (2.20), (2.21) and (2.31), it follows that(Es

1u

Es1v

)= Γ s

1R1Γ1

(Ei

0u

Ei0v

), (2.32)

where Γ s1 is a rational matrix, given by

Γ s1 =

(us

1 · ur1 us

1 · v1vs1 · ur

1 vs1 · v1

). (2.33)

To obtain the scattering matrix, the incident field must be specified withrespect to the directions parallel and perpendicular to the scattering plane, thatis, the incident field needs to be in the form

Ei0 = Ei

1su(ei0 × vs1) + Ei

1sv vs1. (2.34)

Note that the unit vector ei0× vs1 in eq. (2.34) is parallel to the scattering plane.

The expression in eq. (2.34) for the incident field is related to that in eq. (2.13)as follows: (

Ei0u

Ei0v

)= Γ i

1

(Ei

1su

Ei1sv

), (2.35)

where Γ i1 is a 2-D rotational matrix given by

Γ i1 =

(u0 · (ei0 × vs

1) u0 · vs1

v0 · (ei0 × vs1) v0 · vs

1

). (2.36)

Thus, we can express the scattered field in eq. (2.32) as follows:(Es

1u

Es1v

)= Γ s

1R1Γ1Γi1

(Ei

1su

Ei1sv

). (2.37)

Similarly, for the refracted rays with p = 2, we have(Es

2u

Es2v

)= Γ s

2T2Γ2T1Γ1Γi2

(Ei

2su

Ei2sv

). (2.38)

For the orders p = 3, 4, 5, . . . , we have(Es

pu

Espv

)= Γ s

pTpΓp · · ·R2Γ2T1Γ1Γip

(Ei

psu

Epsv

). (2.39)


The definitions for Γ sp , Tp, Γp, Rp, and Γ i

p are similar to those for the case withp = 1. In this manner, both the incident and scattered electric field vectorsare expressed with respect to the scattering plane in eqs (2.37)–(2.39). Thus,the contributions of the emerging or scattered rays to the amplitude scatteringmatrix can be expressed in the forms

A(1) =

(A

(1)2 A

(1)3

A(1)4 A

(1)1

)= Γ s

1R1Γ1Γi1, for externally reflected rays, (2.40)

A(2) =

(A

(2)2 A

(2)3

A(2)4 A

(2)1

)=

Γ s2T2Γ2T1Γ1Γ

i2,

for second-order transmitted rays,(2.41)

A(3) =

(A

(3)2 A

(3)3

A(3)4 A

(3)1

)=

Γ spTpΓp · · ·R2Γ2T1Γ1Γ

ip,

for pth (p > 2)-order transmitted rays.(2.42)

In the foregoing discussion, we have not accounted for the phase change asso-ciated with the optical paths of rays. The rays incident on the scattering particleat different locations must experience phase change due to different paths. Notic-ing this feature, Cai and Liou (1982) considered the phase interference of theemerging rays in ray-tracing computations. Most follow-on studies reported inthe literature essentially ignored the phase shifts associated with raypaths. How-ever, it should be pointed out that the approach developed by Stamnes and Heier(1998) and Chen and Stamnes (1998) can also be used to effectively account forthe phase inference of rays. Takano and Jayaweera (1985) showed that the phaseinterference can be smoothed out when ice crystals are randomly oriented. Forpractical applications, we normally assume the random orientation condition forice crystals in radiative transfer computations. If ice crystals are horizontallyoriented, the single-scattering properties depend not only on the scattering an-gle but also on the azimuth of the scattering plane. In this case, the radiativetransfer calculation can be quite involved (Takano and Liou, 1989b). In additionto random orientation, the integration over the size spectrum will smooth outthe fluctuations produced by phase interferences in the scattering pattern forone size.

For randomly oriented particles, the corresponding phase matrix has onlysix independent elements (van de Hulst, 1957). Thus, for the pth-order emergingray, its contribution to the phase matrix is given by the following expression(Takano and Jayaweera, 1985):

F (p) =

(M(p)1 + M

(p)2 + M

(p)3 + M

(p)4 )/2 (M(p)

2 − M(p)1 )/2 0 0

(M(p)2 − M

(p)1 )/2 (M(p)

2 − M(p)3 − M

(p)4 + M

(p)1 )/2 0 0

0 0 S(p)12 + S

(p)34 −D

(p)12

0 0 D(p)21 S

(p)12 − S

(p)34

,

(2.43)

where the matrix F (p) transforms the incident Stokes parameters to the scatteredStokes parameters associated with the pth-order outgoing localized wave. In eq.(2.43), the phase matrix elements are defined by


M(p)i = |A(p)

i |2, (2.44)

S(p)ij = S

(p)ji = (A(p)

I A(p)∗j +A

(p)∗i A

(p)j )/2, (2.45)

D(p)ij = −D(p)

ji =√−1(A(p)

i A(p)∗j −A

(p)i A

(p)j )/2, (2.46)

where the subscripts i and j range from 1 to 4, and the asterisk indicates complexconjugate. Thus, the scattering matrix associated with the various orders ofexternal reflections and transmissions of all the incident rays is given by

Fray =N∑

j=1

∞∑p=1

∆σjF(p)j /

∞∑p=1

∆σj , (2.47)

where j denotes that the external reflection and the various orders of transmis-sion are associated with the jth initial ray, N is the total number of incidentrays, and ∆σj is the cross-section of the jth initial ray. To speed up the com-putation, the foregoing ray-tracing algorithm can be implemented by using theMonte Carlo method. Interested readers may wish to consult with the papers byWendling et al. (1979), Takano and Liou (1995), Macke (1993), and Yang andLiou (1998). More recently, an efficient algorithm for specifying the incident raysin the Monte Carlo ray-tracing technique implemented for convex geometries hasbeen reported by Zhang et al. (2004).

In addition to the contributions from the reflected and refracted rays, diffrac-tion also contributes to the scattering of the incident wave. According to Babi-net’s principle (Born and Wolf, 1959), the diffraction pattern associated with anobject is the same as that for an aperture with a shape identical to the projectionof the object on a plane normal to the incident direction. The diffraction matrixobtained by the scalar Fraunhofer diffraction theory for a scattering particle hasbeen extensively employed in the previous ray-tracing studies. Yang and Liou(1998) showed that the scattering matrix associated with diffraction is given inthe form

Adif =k2

2πD

[(cos θ + cos2 θ)/2 0

0 (1 + cos θ)/2

], (2.48)

whereD =

∫∫projected

area

exp(−ikr · ξ) d2ξ. (2.49)

From eqs (2.48) and (2.49), the contribution of diffraction to the scatteringphase matrix, denoted as Fdif , can be evaluated. To sum the contributions dueto diffraction and Fresnel rays, proper weighting factors must be accounted for,particularly, in the case when the scattering particle is absorptive.

If the scattering ice crystal is absorptive, i.e., the imaginary refractive indexis nonzero, the total absorption can be accounted for by considering the absorp-tion of individual rays. In general, the absorption cross of a particle dependson the polarization configuration of the incident light. However, for randomlyoriented particles, their absorption cross-section is the average of the absorption


cross-sections for two orthogonal polarization cases. Consider a case where thepolarization of the incident light is specified as follows:

(Ei0u,E

i0v) = (1, 0). (2.50)

The intensity of the first-order refracted field can be obtained from eq. (2.26) inthe form

(Et0u,E

t0v)(Et

0u,Et0v)∗,+

∣∣∣(Ei

0u,Ei0v)=(1,0)

= (1, 0)(T1Γ1)+(T1Γ1)∗(1, 0)+, (2.51)

where the superscript symbol + denotes the transpose of a matrix, and ∗ in-dicates the complex conjugate. The intensity given by eq. (2.51) is essentiallythe amplitude of the Poyting vector (Born and Wolf, 1959) in which the re-fractive index and change in the ray cross-section due to refraction have beenimplicitly accounted for in the refractive matrix given by eq. (2.27). A similarexpression can be derived for the case when the polarization of field is given by(Ei

0u,Ei0v) = (0, 1). Therefore, the contribution of the first-order refracted rays

to the absorption cross-section is given by

σabs,1 =N∑

j=1

2−1∆σj [1− exp(−4πmidj1/λ)]

×[(Et

pu,Etpv)(Et

pu,Etpv)

∗,+∣∣(Ei

0u,Ei0v)=(1,0)

+(Etpu,E

tpv)(Et

pu,Etpv)∗,+

∣∣(Ei

0u,Ei0v)=(1,0)

]

=N∑

j=1

2−1∆σj [1− exp(−4πmidj1/λ)]

· [(1, 0)(T1Γ1)+(T1Γ1)∗(1, 0)+ + (0, 1)(T1Γ1)+(T1Γ1)∗(1, 0)+],

(2.52)

where subscript j denotes the jth initial ray, dj1 is the distance between the firstincident point (i.e., Q1 in Fig. 2.1) and the second incident point (i.e., Q2 inFig. 2.1), ∆σj is the cross-section of the jth initial ray, mi is the imaginary partof the refractive index, and λ is the incident wavelength in a vacuum. Likewise,the contribution by the pth-order reflected rays is

σabs,p =N∑

j=1

2−1∆σj [1− exp(−4πmidjp/λ)] exp

(−4πmiλ

−1p−1∑L=1

djL

)

×[(1, 0)(TpΓp · · ·R2Γ2t1Γ1)+(TpΓp · · ·R2Γ2T1Γ1)∗(1, 0)+

+(0, 1)(TpΓp · · ·R2Γ2T1Γ1)+(TpΓp · · ·R2Γ2T1Γ1)∗(0, 1)+], (2.53)


Thus, the absorption cross-section of the scattering particle can be expressed asfollows:

σabs =∞∑

p=1

σabs,p. (2.54)

In practice, the summation in eq. (2.54) can be truncated for the terms withp > 10, because the amount of the energy carried by the higher-order rays is in-significant. Equations (2.52)–(2.54) provide the explicit formulations for absorp-tion cross-section within the framework of the ray-tracing technique in which thepolarization configuration is fully accounted for. In the conventional ray-tracingmethod, the extinction cross-section is twice the projected area of the scatteringparticle, that is

σext = 2σp, (2.55)

where σp is the particle’s projected area on a plane normal to the incidentdirection. The contribution of diffraction to the extinction cross-section is equalto that associated with the externally reflected rays and the transmitted raysthat experience two refractions and various orders of internal reflections.

One of the shortcomings of the conventional ray-tracing method is the pro-duction of the delta-transmission associated with the refraction of rays throughtwo parallel faces of the pristine ice crystals. The delta-transmission phenomenonhas been discussed in detail by Takano and Liou (1989a) and Mishchenko andMacke (1998). Let the portion of the scattering cross-section associated with thedelta-transmission be σδ. Then, the scattering cross-section can be separatedinto three terms as follows:

σsca = (σp − σabs − σδ) + σδ + σp, (2.56)

The first term (σp−σabs−σδ) corresponds to the contribution from the externallyreflected rays and the various transmitted rays excluding the delta-transmittedrays, the second term (σδ) denotes the contribution from the delta-transmittedrays, and the third term (σp) is associated with diffraction. Let fδ be the ratioof the delta-transmitted energy to the total scattered energy defined by

fδ = σδ/σsca = σδ/(2σp − σabs) = σδ/(σext − σabs). (2.57)

Using the standard notation, the scattered Stokes vector can be expressed asfollows:

IsQsUsVs

=

σsca

k2r2P

IiQiUiVi

, (2.58)

where (Ii, Qi, Ui, Vi) and (Is, Qs, Us, Vs) are the incident and scattered Stokesparameters, respectively, and P is the normalized phase matrix. Based on Frayin eq. (2.47), Fdif that is defined on the basis of Adif in eq. (2.48), the expressionsin eqs (2.56) and (2.57), and the associated physical meanings of these quantitiesand expressions, the normalized phase matrix is given by


P (θ) =[(2σp − σabs)(1− fδ)− σp]aFray+2(2σp − σabs)fδδ(cos θ − 1) ¯I+σpbFdif

2σp − σabs

= 2fδδ(cos θ − 1) ¯I +[(1− fδ)− 1

2ωo

]aFray +

12ωo

bFdif , (2.59)

where ω0 = σsca/σext is the single-scattering albedo. In eq. (2.59) θ is the scat-tering angle, ¯I is a unit 4 × 4 matrix, and the two parameters, a and b, arenormalization factors given, respectively, by the following two equations:

a =2∫ π

0 Fray,11(θ) sin θ dθ, (2.60)

b =2∫ π

0 Fdif,11(θ) sin θ dθ, (2.61)

where the subscript 11 indicates the first element of the associated matrix. Withthe normalization factors given in eqs (2.60) and (2.61), it can be shown thatthe phase matrix in eq. (2.59) is normalized in the sense that the first phasematrix element P11 (i.e., the phase function) satisfies the following normalizationcondition:

12

∫ π

0P11(θ) sin θ dθ = 1. (2.62)

The conventional ray-tracing technique utilizes the assumption that the en-ergy attenuated by a scattering particle is equally divided into two parts: ex-tinction associated with diffraction and extinction due to Fresnel reflection andrefraction. In this case, the extinction efficiency (i.e., the ratio of extinctioncross-section to particle projected area) is 2 regardless of the size and shapeof the scattering particle, referred to as the optical theorem. In addition, thecomputation of far field by directly applying the ray-tracing technique leads tothe delta-transmission (Takano and Liou, 1989a) in the forward direction, as isevident from the presence of a delta function in eq. (2.59). To overcome theseshortcomings, Yang and Liou (1995, 1996a, 1996b, 1997) have developed twoimproved geometric optics methods.

According to the fundamental theory of classic electrodynamics, the far fieldcan be exactly computed if the tangential components of the electric and mag-netic fields on the surface of a scattering particle are known (Jackson, 1998,p. 485) in the form

Es(r)∣∣kr→∞ =

eikr

−ikrk2

4πr ×∮∫{ns ×E(r′)− r × [ns ×H(r′)]} e−ikr·r′

d2r,

(2.63)where the integral domain is the surface of the scattering particle. In essence,eq. (2.63) is derived on the basis of the electromagnetic equivalence theorem(Schelkunoff, 1943). Thus, the geometric ray-tracing technique can be used tocompute the near field on the particle’s surface, which can be subsequentlymapped to the far field on the basis of eq. (2.63). The mapping idea withinthe framework of the ray-tracing computation was first employed by Muinonen


(1989) who developed an algorithm known as the modified Kirchhoff approxima-tion (MKA) to solve the scattering by nonspherical ice crystals based on an elec-tromagnetic equation similar to eq. (2.63). To simplify numerical computations,a constant extinction efficiency of 2 was assumed in the MKA and the strongforward-scattering maximum was approximated by the Fraunhofer diffraction.Yang and Liou (1995, 1996b) considered the phase interference and the polar-ization state of the rays in the near-field computation (hereafter, this method isreferred to as GOM2). In GOM2, the extinction cross-section is computed fromthe fundamental extinction formula (van de Hulst, 1957) or the so-called opticaltheorem (Bohren and Huffman, 1983) given by

σext =2πk2 Re

[A‖(ei0) +A⊥(ei0)

], (2.64)

where k = 2π/λ in which λ the incident wavelength. A‖(ei0) and A⊥(ei0) arethe amplitude scattering matrix elements in the forward scattering direction forparallel and perpendicular polarization configurations, respectively. In eq. (2.64),the symbol, Re, denotes the real part of the associated quantity. In GOM2, theabsorption cross-section of an ice crystal is computed via a rigorous electromag-netic relation (Hage et al., 1991) given by

σabs =kεi|Ei|2

∫∫∫

v

E(r) ·E∗(r dr, (2.65)

where εi is the imaginary part of the permittivity, Ei is the incident electricvector, and the integration is carried out for the volume of the particle. Wheneq. (2.65) is applied to the ray-tracing computation, a semi-analytical expressioncan be derived for the absorption cross-section, as shown by Yang and Liou(1996b).

Similar to the case in eq. (2.63), the far field can be computed if the internalelectric field within the scattering particle is known, as is given by the followingelectrodynamic relation (Saxon, 1973; Goedecke and O’Brien, 1988; Mishchenkoet al., 2002):

Es(r)∣∣kr→∞ =

k2 eikr

4πr(ε− 1)

∫∫∫

v

{E(r′)− r[r ·E(r′)]} e−ikr·r′d3r′, (2.66)

where ε is the permittivity. The advantage of using eq. (2.66) is that only theelectric field is involved, whereas both electric and magnetic fields are includedin eq. (2.64). In the limit of geometric optics, the incident wave consists of abundle of rays each of which propagates along a rectilinear path determinedby Snell’s law. With this assumption, the volume integration in eq. (2.66) canbe carried out along the ray paths, and at the same time a semi-analyticalsolution for the far-field can be derived (Yang and Liou, 1997). This method isreferred to as the ray-by-ray integration (RBRI) method in the authors’ previouswork. Yang and Liou (1997) also showed that RBRI reduces to the anomalousdiffraction approximation (ADA) developed by van de Hulst (1957) when thescattering particle is optically tenuous (i.e., the refractive index is close to 1).


The GOM2 and RBRI methods are essentially the same within the context that ahybrid algorithm based on the principles of geometric optics and the near-to-far-field electromagnetic wave theory are employed to compute the single-scatteringproperties of an ice crystal.

2.3 The finite-Difference Time Domain Method

For size parameters less than about 20–40, the geometric optics method breaksdown (Yang and Liou, 1995; 1996b; Macke et al., 1995; Mishchenko and Macke,1999). Although various methods (see Mishchenko et al. 2000) have been devel-oped to solve for the single-scattering properties of nonspherical particles, thefinite-difference time-domain (FDTD) method pioneered by Yee (1966) is quiteattractive for the computation of light scattering and absorption by small non-spherical and inhomogeneous ice crystals. The FDTD method has been known tobe a flexible and robust approach for solving various electromagnetic problems.Publications related to the FDTD method in the literature surveyed by Shlagerand Schneider (1998) illustrate the popularity of this method. In particular, sev-eral books have been entirely devoted to this particular numerical technique forvarious applications, ranging from the signal propagation in circuits to the studyof the electromagnetic hazard in bioscience (Kunz and Luebbers, 1993; Taflove,1995; Taflove and Hagness, 2000).

The FDTD method can be technically considered as a ‘brute force’ approachto solve the time-dependent Maxwell curl equations. Unlike the conventionalapproach of solving Maxwell’s equations in the frequency domain in which anelectromagnetic scattering process is posed as a boundary-value problem, theFDTD method solves an electromagnetic problem as an initial-value problem.Mathematically, a boundary-value problem is normally more difficult than itsinitial-value counterpart. To illustrate the basic concept of the FDTD method forthose who do not have any experience on this numerical technique, here we firstrecapture the FDTD solution for the propagation of a plane electromagnetic wavein free space, which is a typical 1-D wave-propagation problem. Then, we outlinethe other major numerical aspects (namely, the absorbing boundary condition,the transform of the field from the time domain to the frequency domain, and themapping of the near field to the far field) involved in the implementation of theFDTD technique for computing the scattering properties of dielectric particles.A comprehensive discussion on the 1-D scalar wave equation in the frameworkof the finite-difference technique has been presented by Umashankar and Taflove(1982), Taflove (1995) and Taflove and Hagness (2000) who also discussed the1-D electromagnetic wave propagation for the implementation of the incidentwave condition in the FDTD numerical computation.

As shown in Fig. 2.3, a wave propagates along the z-axis of a Cartesiancoordinate system. The electric and magnetic vectors associated with this waveare specified along the x- and y-axis of the coordinate system. This is a typical 1-D wave propagation problem and the governing equations for the electromagneticwave can be written as follows:


Fig. 2.3. The grid for the one-dimensional finite-difference analog of Maxwell’s equa-tions. The propagation of a plane electromagnetic wave is defined as propagating alongthe z-axis of the coordinate system. The electric and magnetic fields are specified ina staggered manner, i.e., Ex is defined at grid points k = 0, 1, 2, . . . , K whereas Hy

specified at grid points k = 1/2, 3/2, . . . , k − 1/2.

1c

∂Ex(z, t)∂t

= −∂Hy(z, t)∂z

, (2.67)

−1c

∂Hy(z, t)∂t

=∂Ex(z, t)

∂z, (2.68)

where c is the speed of light in vacuum. To solve variation in the electromagneticwave, the finite-difference technique is used in the FDTD method, i.e., bothelectric and magnetic fields are specified in terms of their discrete values in timeand space. Following Yee (1966) and Taflove (1995), we define the discrete valuesof the fields as follows:

Enx,k = Ex(k∆z, n∆t), (2.69)

Hn+1/2y,k+1/2 = Hy [(k + 1/2) ∆z, (n+ 1/2) ∆t] , (2.70)

where ∆z and ∆t are the grid size and time increment, respectively, and theindices k and n are integers. The electric and magnetic fields are defined on astagger grid, i.e., the electric field is specified at grid points with integer indices(k = 0, 1, 2, . . . ), whereas the magnetic field is defined at the middle points ofadjacent grid points. Similarly, the electric field is defined at time steps n∆t,whereas the magnetic field is defined at time steps (n+1/2) ∆t. Using the discretevalues of the fields defined in eqs (2.69) and (2.70), the derivatives of the electricand magnetic fields in Maxwell’s equations can be expressed in terms of the


standard ‘leapfrog’ or central difference scheme as follows (Yee, 1966; Taflove,1995):

∂Ex(z, t)∂t

∣∣∣∣z=k∆zt=(n+1/2)∆t

≈ Ex[k∆z, (n+ 1) ∆t]− Ex[k∆z, n∆t]∆t

=En+1

x,k − Enx,k

∆t, (2.71)

∂Hy(z, t)∂z

∣∣∣∣z=k∆zt=(n+1/2)∆t

≈ Hy[(k + 1/2) ∆z, (n+ 1/2) ∆t]−Hy[(k − 1/2) ∆z, (n+ 1/2) ∆t]∆z

=H

n+1/2y,k+1/2 −H

n+1/2y,k−1/2

∆z, (2.72)

Upon substitution of eqs (2.71) and (2.72) into eq. (2.67), the finite-differenceanalog of eq. (2.67) is given by

En+1x,k = En

x,k +c∆t

∆z

[H

n+1/2y,k−1/2 −H

n+1/2y,k+1/2

]. (2.73)

Similarly, we can derive the finite difference analog of eq. (2.68) as follows:

Hn+1/2y,k+1/2 = H

n−1/2y,k+1/2 +

c∆t

∆z

[En

x,k − Enx,k+1

]. (2.74)

The selection of the time increment ∆t in the finite analog of Maxwell’s equationsis not arbitrary because of the stability condition required by the differenceequations in numerical computation (Yee, 1966; Taflove and Brodwin, 1975).In the case for 1-D electromagnetic wave, the stability condition requires thatc∆t/∆z ≤ 1.

Equations (2.73) and (2.74) constitute the difference equations for comput-ing variation in the fields. In practice, if the initial values of the fields, say, E1

x,k

and H1/2y,k+1/2 are defined at grid points indicated by indices 0, 1, 2, . . . and 1/2,

3/2, 5/2, . . . , respectively, this variation can be simulated by a time-marchingiteration based on eqs (2.73) and (2.74). However, the computational domain inthe numerical simulation must be truncated. For the 1-D case, let the computa-tional domain be the spatial regime bounded by grid point k = 0 and k = K.From eqs (2.73) and (2.74), the electric fields at k = 0 and k = K cannot becomputed by these two finite difference equations because H1/2

y,−1/2 and H1/2y,K+1/2

are defined at the locations outside the computational domain. Thus, to updatethe electric fields at the boundary points, appropriate boundary conditions mustbe provided. For simplicity in calculating the field values at the boundary gridpoints, the grid configuration can be specified as follows:


c∆t/∆z = 1/2. (2.75)

The preceding relation implies that the wave propagates half grid size every timestep. Thus, the equations for updating the boundary values can be written asfollows:

Enx,1/2 = En−1

x,1 , (2.76)

En+1x,0 = En

x,1/2, (2.77)

Enx,K−1/2 = En−1

x,K−1, (2.78)

En+1x,K = En

x,K−1/2, (2.79)

where Enx,1/2 and En

x,K−1/2 are two auxiliary quantities introduced for updatingthe electric fields at the two boundary points, respectively. Equations (2.76) and(2.77) constitute the boundary condition for the grid point for k = 0, whereaseqs (2.78) and (2.79) constitute the boundary condition for the grid point ofk = K. Using the finite difference equations (2.73) and (2.74) and the bound-ary conditions given by eqs (2.76)–(2.79), one can compute variation in theelectromagnetic fields within the region between k = 0 and K. The computedelectromagnetic fields are the same as those in the case where the computationaldomain is not bounded. Note that if En

x,0 = 0 and Enx,K = 0, the boundary

grid points constitute the reflecting boundaries. In this case, the electromag-netic wave, when impinges on the boundary grid points, is reflected back to thecomputational domain and contaminate the numerical simulation. The preced-ing 1-D finite-difference equations and the associated boundary conditions havebeen employed to implement the incident wave conditions in various applicationsof the FDTD technique (e.g., Sullivan et al., 1988).

The discretization of Maxwell’s equations in the 3-D case is similar to thatin 1-D case. For example, the Maxwell curl equations in the scalar componentform for Ex and Hx can be written as follows:

∂Ex(r, t)∂t

=c

ε

(∂Hz(r, t)

∂y− ∂Hy(r, t)

∂z

), (2.80)

∂Hx(r, t)∂t

= −c(∂Ez(r, t)

∂y− ∂Ey(r, t)

∂z

). (2.81)

For simplicity in this discussion, we assume that the permittivity is a real num-ber. Using the standard definitions for the electric and magnetic fields on a gridcell shown in Fig. 2.4 (Yee, 1966), it is straightforward to derive the differenceanalog of eqs (2.81) and (2.82) on the basis of the central-difference scheme inboth time and space as follows:


Fig. 2.4. Locations of the electric and magnetic field components on a cubic grid cell,defined by Yee (1966).

En+1x,i,j+1/2,k+1/2 = En

x,i,j+1/2,k+1/2

+1

εi,j+1/2,k+1/2

[c∆t

∆y(Hn+1/2

z,i,j+1,k+1/2 −Hn+1/2z,i,j,k+1/2)

+c∆t

∆z(Hn+1/2

y,i,j+1/2,k −Hn+1/2y,i,j+1/2,k+1)

], (2.82)

Hn+1/2x,i+1/2,j,k = H

n−1/2x,i+1/2,j,k +

[c∆t

∆y(En

z,i+1/2,j−1/2,k − Enz,i+1/2,j+1/2,k)

+c∆t

∆z(En

y,i+1/2,j,k+1/2 − Eny,i+1/2,j,k−1/2)

]. (2.83)

The finite-difference equations for other components of the electric and magneticfields can be determined in similar forms. In the 3-D case, the stability conditionhas been derived by Taflove and Brodwin (1975) given by

c∆t ≤ 1√1/∆x2 + 1/∆y2 + 1/∆z2

. (2.84)

In the 3-D case, the boundary condition is a major issue in the FDTD com-putation because the analytical boundary equations cannot be constructed dueto the unknown propagating directions of the outgoing waves. In the past twodecades, various numerical techniques (Blaschak and Kriegsmann, 1988; Mooreet al., 1988; Berntsen and Hornsleth, 1994) have been developed to update thefield values at the boundary grid points. The commonly used approaches arethe absorbing boundary condition developed by Mur (1981), the transmittingboundary condition developed by Liao et al. (1984), and the perfectly matchedlayer (PML) boundary condition pioneered by Berenger (1994, 1996). The PML


is particularly efficient and popular in the implementation of the FDTD nu-merical scheme (e.g., Katz et al. 1994). In the following we briefly outline theprinciple of the PML boundary condition.

The essence of PML is to introduce an artificial absorbing medium withinthe boundary layers that impose absorption on the partial electromagnetic fieldcomponents. For implementing the PML absorbing boundary condition, a fieldcomponent needs to be split into two parts in the forms

(Ex, Ey, Ez) = [(Exy + Exz), (Eyx + Eyz), (Ezx + Ezy)], (2.85)(Hx, Hy, Hz) = [(Hxy + Exz), (Hyx +Hyz), (Hzx +Hzy)]. (2.86)

The PML boundary condition equations for Ez and Hz components at a bound-ary perpendicular to the x-axis are given by

exp[−τx(x)t]c

∂

∂t{exp[τx(x)t]Ezx} =

∂(Hyx +Hyy)∂x

, (2.87)

1c

∂Ezy

∂t= −∂(Hxy +Hxz)

∂y, (2.88)

exp[−τx(x)t]c

∂

∂t{exp[τx(x)t]Hzx} = −∂(Eyx + Eyz)

∂x, (2.89)

1c

∂Hzy

∂t=

∂(Exy + Exz)∂z

, (2.90)

where τx(x) is defined for the boundary layers near x = 0 as follows:

τx(x) = τmax|(x− δx)/δx|p, (2.91)

where τmax denotes the maximum absorption at x = 0, which can be determinedby specifying the reflectance of the boundary layers at a normal incidence. In eq.(2.91), δx denotes the thickness of the boundary layer and p is usually selectedbetween 2 and 3 (Lazzi and Gandi, 1996). In numerical computations, the PMLboundary condition is applied to outgoing scattered waves. To do so, the FDTDcomputational is usually divided into an inner domain enclosed by an outerdomain. Within the inner domain, the total field (incident plus scattered fields)is simulated, and only outgoing or scattered field is simulated otherwise. Inpractice, an interface known as the Huygens surface (Merewether et al., 1980)is introduced to connect the fields in the two domains. Note that, to avoidthe Huygens surface, an alternative approach for which the scattered field isdefined for the entire computational domain has been developed (e.g., Britt,1989; Yang and Liou, 1995). However, this approach is not computationallyefficient, particularly, for the implementation of the FDTD method in the 3-Dcase.

With the finite difference analog of the Maxwell equations and the absorbingboundary condition, the electromagnetic field within or near a scattering parti-cle (i.e., the near field) in the time domain can be obtained. The correspondingsignal in the frequency domain can be obtained via the discrete Fourier trans-form. For example, if the time series of electric field at a given grid point say,


En+1x,i,j+1/2,k+1/2, are known, the corresponding signal in the frequency domain

can be obtained as follows

Ex,i,j+1/2,k+1/2(k) =N∑

n=0

En+1x,i,j+1/2,k+1/2 exp(i2πcn∆t/λ), (2.92)

The discrete Fourier transform is more efficient than the fast Fourier transformfor application to the FDTD computation, as illustrated by Furse and Gandhi(1995).

After the near field in the frequency domain is obtained, the scattered farfield can be obtained from either eq. (2.63) or eq. (2.66). Although these twoequations are physically equivalent, the far-field values computed from eqs (2.63)and (2.66) differ in terms of accuracy (Zhai et al., 2004). The near-to-far-fieldtransformation in both the frequency and time domains has been discussed byTaflove and Hagness (2000) and references cited therein.

2.4 Numerical Examples

The FDTD technique has been applied to the investigation of the scatteringproperties of small ice crystals (Yang and Liou, 1995, 1996a; Sun et al., 1999;Sun and Fu, 2000; Yang et al., 2004a) and its accuracy has been extensivelystudied in reference to the results computed from the ‘exact’ Lorenz–Mie theoryfor spheres. Also, Baran et al. (2001) compared the FDTD solution and T-matrix results for the single-scattering properties of hexagonal ice crystals. Ingeneral, the relative errors of the FDTD solutions are typically less than 1%for computing the extinction and absorption cross-sections and on the order of10% for computing the phase function if the grid size is less than 1/20 of thewavelength within the scattering particle of interest. The accuracy of the FDTDsolution can be further improved if a finer grid size is used, but at the expense ofthe computational CPU time. In practice, application of the FDTD technique tothe light scattering by ice crystals is limited to size parameters less than about 20because enormous computational efforts are required. Sun et al. (1999) appliedthis method to the solution to the scattering of light by a single ice sphere witha size parameter of 40 at a wavelength at which the refractive index of ice issmall. Application of the FDTD technique to the scattering of light by particleswith a large refractive index (e.g., the real part of the refractive index is on theorder of 8) has been recently reported by Sun and Fu (2000) and Yang et al.(2004a). The latter authors also applied this method to complex bullet rosette icecrystals with various branches (Yang et al., 2004a). As an example, Fig. 2.5 showscomparison of the phase matrix elements computed from the FDTD techniqueand the Lorenz–Mie theory for an ice sphere with a size parameter of 20 at awavelength of 0.6328 µm. A surface-integral based approach (Zhai et al., 2004)with a grid resolution of λ/∆s = 40 is used to map the near field to far fieldfor the FDTD resolute shown in Fig. 2.5, where λ and ∆s are the incidentwavelength and grid size, respectively. Excellent agreement between the FDTD


Scattering Angle ( o )

1x10-3

1x10-2

1x10-1

1x100

1x101

1x102

1x103

0 30 60 90 120 150 180

Lorenz-Mie

FDTD

Sphereka=20λ=0.6328 µm

m=1.308+i1.09x10-8

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

P43/P11

P33 /P11

P12/P11

Fig. 2.5. Comparison of the phase matrix elements computed from the Lorenz–Mietheory and the FDTD method for an ice sphere with a size parameter of 20 using awavelength of 0.6328 µm.

and Lorenz–Mie results is evident. The FDTD solution for the phase function ismore accurate than for the other phase matrix elements because former is lesssensitive to phase variation in the scattered waves.

It has been commonly assumed that a small quasi-spherical ice crystal maybe approximated by an equivalent sphere, defined by (1) the same diameter(D), (2) the same surface area (A), (3) the same volume (V ), or (4) the sameratio of V to A. Fig. 2.6 shows comparison of the phase functions computedfor these four definitions of spherical equivalence for Platonic solids (i.e., tetra-hedron, hexahedron, octahedron, dodecahedron, and icosahedron). It is inter-esting to note that the study of the Platonic-solid shape has a rich history,which goes back to the beginning of recorded human civilization. For example,


Fig. 2.6. The phase function of an ice sphere with a size parameter of x = 5. Thewavelength and refractive index are 0.6328 µm and 1.3085 + il.09 × 10−8, respectively.Also shown are the phase functions of the five Platonic shapes with the same radius(the first column), projected area (the second column), volume (the third column) andV/A (the fourth column) as those for the ice sphere (after Yang et al., 2004b).

the polyhedron was extensively used/investigated in ancient Egyptian, Baby-lonian, Chinese, and Greek cultures associated with the study of architecture,art, mathematics, and even the philosophy regarding the early understanding


of the universe. Historically, it was believed that matter was composed of afew elemental substances combined in different ways. Influenced by Aristotle(384–322 BC), ancient wisdom assumed that the basic elements (fire, air, earth,water, and celestial matter) were related to the five regular polyhedra. A re-view of the history of the study of polyhedron can be found in a monographby Cromwell (l997) and also in a review article by Field (1979). Although thepolyhedron has been studied for millennia from various perspectives, it is still achallenging topic for modern mathematicians. In fact, many mathematical theo-rems related to the polyhedron have only recently been proved (e.g., Grunbaum,1967). The numbers of the faces for a tetrahedron, a cube, an octahedron, adodecahedron, and an icosahedron are 4, 6, 8, 12, and 20, respectively. The facesof a Platonic solid are equilateral polygons with the same number of sides. Thenumber of faces, the number of vertices, and the number of edges of a polyhe-dron satisfy the famous Euler’s theorem (Euler, 1758) that can be expressed asfollows:

f + v + e = 2, (2.93)

where f is the number of polygon faces, v the number of vertices, and e thenumber of edges. The five platonic solids approach spheres in an orderly man-ner and, therefore, they are ideal for investigating the asphericity effect on thescattering properties. It is evident from Fig. 2.6 that a systematically optimizeddefinition for ‘spherical equivalence’ does not exist. It is seen that the ‘sphericalequivalence’ based on the particle dimension leads to the best approximation inthe case for dodecahedron, whereas the volume-based ‘equivalence’ is more ac-curate than the other three definitions in the case of the icosahedron. It is clearthat the extent of nonsphericity of a particle in the context of light scatteringcomputation depends on a specific geometry. This implies that a general opti-mal ‘spherical equivalence’ cannot be defined to minimize the errors associatedwith the spherical approximation for a variety of nonspherical geometries in lightscattering computations.

The simplest ice crystal shape can be represented by the pristine columnand plate that normally have a basic hexagonal symmetric structure. Using thegeometric ray-tracing technique, the scattering properties of hexagonal ice crys-tals have been extensively investigated in the past. The pristine ice crystal typesproduce the well-known 22◦ and 46◦ halos, as well as a number of fascinatingarcs and sundogs that have been observed in cirrus cloud conditions (see forexample, Greenler, 1990).

The upper panels of Fig. 2.7 show the nonzero phase matrix elements forrandomly oriented hexagonal ice columns at a wavelength of 0.6328 µm. For thescattering of light by large particles with a size parameter on the order of thoseshown in Fig. 2.7, the conventional and improved geometric optics methodsproduce about the same results. The peaks at 22◦ and 46◦ scattering angles inthe phase function are responsible for the halos observed in the atmosphere. Thescattering maximum between 150◦ and 160◦ is produced by the rays undergoingtwo internal reflections (Takano and Liou, 1989a). The lower panels in Fig. 2.7


Fig. 2.7. The scattering phase matrix of randomly oriented hexagonal ice columnsand droxtal ice crystals computed from the geometric optics method for large sizeparameters.


show the nonzero phase matrix elements for randomly oriented droxtal, an icecrystal term introduced by Thuman and Robinson (1954) and Ohtake (1970).The droxtal geometry with a 20-face structure has been suggested as a betterrepresentation of small quasi-spherical ice crystals observed in ice clouds. It hasbeen speculated that the formation of droxtal ice crystals is associated with thefreezing of supercooled water droplets and subsequent growth by water vapordeposition (Zhang et al., 2004). From Fig. 2.7, it is evident that the single-scattering properties of droxtal ice crystals are substantially different from thoseof well-defined pristine hexagonal ice crystals. For the former, the phase functionis quite flat at large scattering angles from 100◦ to 180◦. Additionally, droxtalsscatter less energy than hexagonal ice crystals in the scattering directions around60◦. A strong peak at the 11◦ scattering angle in the phase function of droxtals isproduced by rays undergoing two sequential refractions through the trapezoidaland rectangular faces.

Fig. 2.8 illustrates the phase matrix for randomly oriented small hexagonalice crystals and droxtals computed from FDTD. Based on laboratory and aircraftobservations, small ice crystals tend to have unit aspect ratio (Auer and Veal,1970), i.e., L/2a ∼ 1 in which L and a are the length and semi-width of an icecrystal, respectively. The pronounced scattering peaks corresponding to halos arenot observed in the phase function. However, a scattering maximum is shownfor both small hexagons and droxtals that are randomly oriented in space. Wealso note that the phase function for droxtals shows fluctuations in the sideand backscattering directions, which cannot be smoothed out through randomorientation averaging. Although the overall geometry of a droxtal is close toa sphere, significant nonsphericity effect is noted from the phase matrix. Fora sphere, the ratio P22/P11 is one. It has been argued that the deviation ofP22/P11 from unity is an index of nonsphericity effect (Bohren and Huffman,1983; Mishchenko et al. 2002). From Fig. 2.8, the P22/P11 values for droxtalsfor the scattering angles larger than 60◦ are substantially deviated from unity,indicating the prominent nonspherecity effect. Yang et al. (2003) and Zhang etal. (2004) have proposed that small quasi-spherical ice crystals in ice clouds maybe approximated as droxtals in light scattering computations.

To compare the single-scattering properties for various ice crystal shapes,Figs. 2.9 and 2.10 show the phase functions for six ice crystal habits with smalland large size parameters, respectively. For large aggregates, their surfaces areassumed to be moderately rough in the phase function computation on the basisof the Gram–Charlier distribution (Cox and Munk, 1954) following the methoddescribed in Yang and Liou (1998). It is evident from Fig. 9 that the scatteringof light by small ice crystals does not produce halo peaks. For small platesand columns (the panels in the second row) the phase functions are smoothfor scattering angles from 90◦ to 180◦. On the contrary, the overall feature ofthe phase function for large ice crystals illustrates pronounced peaks, exceptin the case of aggregates because surface roughness smooths out the scatteringpeak.


Scattering Angle ( o )

1x10-2

1x10-1

1x100

1x101

1x102

0 30 60 90 120 150 180

DroxtalkR=10

λ=0.6328 µm

m=1.308+i1.09x10-8

-1

0

1

0.5

0.75

1

-1

0

1

-0.50

0.51

-1

0

1

0 30 60 90 120 150 180

-P12

/P11

P22

/P11

P33/P11

-P43/P11

P44/P11

1x10-2

1x10-1

1x100

1x101

1x102

0 30 60 90 120 150 180

Hexagonal columnka=10, L/2a=1 µm

λ=0.6328 µm

m=1.308+i1.09x10-8

-0.5

0

0.5

0

0.5

1

-1

0

1

-0.5

0

0.5

-1

0

1

0 30 60 90 120 150 180

-P12

/P11

P22

/P11

P33/P11

-P43/P11

P44/P11

Fig. 2.8. The phase matrix of small compact hexagonal and droxtal ice crystals com-puted from the FDTD method.


Fig. 2.9. Comparison of the phase functions computed from the FDTD method forice ice crystal shapes that are commonly observed in ice clouds. The parameter, D,is the maximum dimension for a droxtal, a bullet rosette, or an aggregate ice crystal.For plates and columns, a denotes the half-width and L is the length (for columns) orthickness (for plates). K = 2π/λ is the wavenumber.

To demonstrate the improvement in GOM2 (Yang and Liou, 1995) as com-pared to the conventional ray-tracing approach for moderate size parameters,Fig. 2.11 shows the phase function computed by the two methods and FDTD ina 2D case. In the computation, ice crystals were assumed to be infinitely longhexagonal columns with normal incidence. Halo peaks are noticed in the conven-tional ray-tracing solution, but not in the FDTD result. The scattering patternsproduced by the improved geometric optics are similar to those shown in FDTDfor both size parameters, but its accuracy is degraded in scattering angles largerthan ∼100◦.

Fig. 2.12 shows the extinction efficiency and single-scattering albedo com-puted from FDTD, GOM2 based on eq. (2.63), RBRI based on eq. (2.66), andthe conventional ray-tracing technique. The limitation of the conventional ray-tracing method is evident in the evaluation of the extinction efficiency, which isequal to 2 regardless of the size parameter. At a size parameter of about 20, theresults computed from RBRI, GOM2, and FDTD converge. Owing to the limi-tations of the geometric optics approximation and computational requirements


Fig. 2.10. Comparison of the phase functions computed from the geometric opticsmethod for six ice crystal shapes. For aggregates, surface roughness is included in thelight scattering computation.

in the FDTD method, they can be applied to large (>20) and small (<20) sizeparameters, respectively. However, by combining GOM2 and FDTD, calculationof the single-scattering properties for various ice crystal shapes and sizes can becarried out. This is the essence of the unified theory concept developed by Liouet al. (2000) in the sense that the accurate FDTD solution can be used for smallsize parameters and at the same time an approximate geometric optics approachcan be applied to large size parameters.

2.5 Summary

In this chapter, we have reviewed the theoretical development and numericalcomputation for the single-scattering properties of atmospheric ice crystals. Ap-plication of two numerical methods, the geometric optics approach and theFDTD technique, to the scattering of light by ice crystals have been highlighted.Specifically, we recaptured the ray-tracing methodology originally developed byCai and Liou (1982), which is systematically formulated in a vector form in thepresent presentation. The vector formulation of the ray-tracing procedure is in-dependent of specific coordinate systems and can be implemented in numerical


Fig. 2.11. Normalized phase functions computed by the FDTD, conventional ray-tracing, and GOM2 methods for the scattering of light by randomly oriented 2-Dhexagonal ice crystals (after Yang and Liou, 1995).

computations effectively. The weightings of diffraction, Fresnel refraction andreflection, and the delta-transmission have been explicitly given in the formula-tion of the phase matrix. Moreover, the absorption cross-section of ice crystalsunder the randomly oriented condition is also presented within the frameworkof the geometric optics approach in which both polarization configurations areaccounted for.


Fig. 2.12. Comparison of the extinction efficiency and the single-scattering albedocomputed by FDTD, RBRI, GOM2, and the conventional ray-tracing methods at the0.55-µm and 3.7-µm wavelengths (data taken from Yang and Liou, 1997).

The FDTD techniques pioneered by Yee (1966) and further developed bymany others (e.g., Taflove and Hagness (2000) and references cited therein) isan attractive approach to deal with the scattering of light by small ice crystals.We reviewed the basic principle of the FDTD technique by using the 1-D elec-tromagnetic wave propagation process for illustration. The implementation ofFDTD in the 3-D case for the scattering of light by ice crystals was outlined,including the discretization of Maxwell’s equations on the basis of the differenceapproximation, the absorption boundary condition for truncating the computa-tional domain, the transform of simulated signals from the time to the frequencydomain, and the near field to far field mapping.

Finally, selected numerical results were presented to illustrate the scatteringcharacteristics of large and small ice crystals. The overall feature for the scat-tering of light by large ice crystals is that the corresponding phase functionsnormally show strong halo peaks. However, if the surface roughness conditionis imposed in the ray-tracing computation, the scattering peaks associated withhalos are largely smoothed out. For small ice crystals, the scattering phase func-tions are generally featureless in the side and backward scattering directions.


Moreover, we revisited some of our previous numerical results to demonstratecertain advantages of the improved geometric optics methods.

Acknowledgments

The authors thank G. Chen, Z. Zhang and P. Zhai for assisting in numericalcomputation and graphical presentation. During the course of this research, PingYang and K. N. Liou were supported by the National Science Foundation un-der Grants ATM-458131 and ATM-0331550, respectively. Ping Yang would alsolike to acknowledge support (NAG5-11374) from the NASA Radiation SciencesProgram.

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